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 * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
 * version 2 for more details (a copy is included in the LICENSE file that
 * accompanied this code).
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 * 2 along with this work; if not, write to the Free Software Foundation,
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package com.nulldev.util.graphics.renderIt.ginterfaces.marlin.impl.pisces;

final class DCurve {

	double ax, ay, bx, by, cx, cy, dx, dy;
	double dax, day, dbx, dby;

	DCurve() {
	}

	void set(final double[] points, final int type) {
		// if instead of switch (perf + most probable cases first)
		if (type == 8) {
			set(points[0], points[1], points[2], points[3], points[4], points[5], points[6], points[7]);
		} else if (type == 4) {
			set(points[0], points[1], points[2], points[3]);
		} else {
			set(points[0], points[1], points[2], points[3], points[4], points[5]);
		}
	}

	void set(final double x1, final double y1, final double x2, final double y2, final double x3, final double y3, final double x4, final double y4) {
		final double dx32 = 3.0d * (x3 - x2);
		final double dy32 = 3.0d * (y3 - y2);
		final double dx21 = 3.0d * (x2 - x1);
		final double dy21 = 3.0d * (y2 - y1);
		ax = (x4 - x1) - dx32; // A = P3 - P0 - 3 (P2 - P1) = (P3 - P0) + 3 (P1 - P2)
		ay = (y4 - y1) - dy32;
		bx = (dx32 - dx21); // B = 3 (P2 - P1) - 3(P1 - P0) = 3 (P2 + P0) - 6 P1
		by = (dy32 - dy21);
		cx = dx21; // C = 3 (P1 - P0)
		cy = dy21;
		dx = x1; // D = P0
		dy = y1;
		dax = 3.0d * ax;
		day = 3.0d * ay;
		dbx = 2.0d * bx;
		dby = 2.0d * by;
	}

	void set(final double x1, final double y1, final double x2, final double y2, final double x3, final double y3) {
		final double dx21 = (x2 - x1);
		final double dy21 = (y2 - y1);
		ax = 0.0d; // A = 0
		ay = 0.0d;
		bx = (x3 - x2) - dx21; // B = P3 - P0 - 2 P2
		by = (y3 - y2) - dy21;
		cx = 2.0d * dx21; // C = 2 (P2 - P1)
		cy = 2.0d * dy21;
		dx = x1; // D = P1
		dy = y1;
		dax = 0.0d;
		day = 0.0d;
		dbx = 2.0d * bx;
		dby = 2.0d * by;
	}

	void set(final double x1, final double y1, final double x2, final double y2) {
		final double dx21 = (x2 - x1);
		final double dy21 = (y2 - y1);
		ax = 0.0d; // A = 0
		ay = 0.0d;
		bx = 0.0d; // B = 0
		by = 0.0d;
		cx = dx21; // C = (P2 - P1)
		cy = dy21;
		dx = x1; // D = P1
		dy = y1;
		dax = 0.0d;
		day = 0.0d;
		dbx = 0.0d;
		dby = 0.0d;
	}

	int dxRoots(final double[] roots, final int off) {
		return DHelpers.quadraticRoots(dax, dbx, cx, roots, off);
	}

	int dyRoots(final double[] roots, final int off) {
		return DHelpers.quadraticRoots(day, dby, cy, roots, off);
	}

	int infPoints(final double[] pts, final int off) {
		// inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
		// Fortunately, this turns out to be quadratic, so there are at
		// most 2 inflection points.
		final double a = dax * dby - dbx * day;
		final double b = 2.0d * (cy * dax - day * cx);
		final double c = cy * dbx - cx * dby;

		return DHelpers.quadraticRoots(a, b, c, pts, off);
	}

	int xPoints(final double[] ts, final int off, final double x) {
		return DHelpers.cubicRootsInAB(ax, bx, cx, dx - x, ts, off, 0.0d, 1.0d);
	}

	int yPoints(final double[] ts, final int off, final double y) {
		return DHelpers.cubicRootsInAB(ay, by, cy, dy - y, ts, off, 0.0d, 1.0d);
	}

	// finds points where the first and second derivative are
	// perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
	// * is a dot product). Unfortunately, we have to solve a cubic.
	private int perpendiculardfddf(final double[] pts, final int off) {
		assert pts.length >= off + 4;

		// these are the coefficients of some multiple of g(t) (not g(t),
		// because the roots of a polynomial are not changed after multiplication
		// by a constant, and this way we save a few multiplications).
		final double a = 2.0d * (dax * dax + day * day);
		final double b = 3.0d * (dax * dbx + day * dby);
		final double c = 2.0d * (dax * cx + day * cy) + dbx * dbx + dby * dby;
		final double d = dbx * cx + dby * cy;

		return DHelpers.cubicRootsInAB(a, b, c, d, pts, off, 0.0d, 1.0d);
	}

	// Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
	// a variant of the false position algorithm to find the roots. False
	// position requires that 2 initial values x0,x1 be given, and that the
	// function must have opposite signs at those values. To find such
	// values, we need the local extrema of the ROC function, for which we
	// need the roots of its derivative; however, it's harder to find the
	// roots of the derivative in this case than it is to find the roots
	// of the original function. So, we find all points where this curve's
	// first and second derivative are perpendicular, and we pretend these
	// are our local extrema. There are at most 3 of these, so we will check
	// at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
	// points, so roc-w can have at least 6 roots. This shouldn't be a
	// problem for what we're trying to do (draw a nice looking curve).
	int rootsOfROCMinusW(final double[] roots, final int off, final double w2, final double err) {
		// no OOB exception, because by now off<=6, and roots.length >= 10
		assert off <= 6 && roots.length >= 10;

		int ret = off;
		final int end = off + perpendiculardfddf(roots, off);
		roots[end] = 1.0d; // always check interval end points

		double t0 = 0.0d, ft0 = ROCsq(t0) - w2;

		for (int i = off; i <= end; i++) {
			double t1 = roots[i], ft1 = ROCsq(t1) - w2;
			if (ft0 == 0.0d) {
				roots[ret++] = t0;
			} else if (ft1 * ft0 < 0.0d) { // have opposite signs
				// (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
				// ROC(t) >= 0 for all t.
				roots[ret++] = falsePositionROCsqMinusX(t0, t1, w2, err);
			}
			t0 = t1;
			ft0 = ft1;
		}

		return ret - off;
	}

	private static double eliminateInf(final double x) {
		return (x == Double.POSITIVE_INFINITY ? Double.MAX_VALUE : (x == Double.NEGATIVE_INFINITY ? Double.MIN_VALUE : x));
	}

	// A slight modification of the false position algorithm on wikipedia.
	// This only works for the ROCsq-x functions. It might be nice to have
	// the function as an argument, but that would be awkward in java6.
	// TODO: It is something to consider for java8 (or whenever lambda
	// expressions make it into the language), depending on how closures
	// and turn out. Same goes for the newton's method
	// algorithm in DHelpers.java
	private double falsePositionROCsqMinusX(final double t0, final double t1, final double w2, final double err) {
		final int iterLimit = 100;
		int side = 0;
		double t = t1, ft = eliminateInf(ROCsq(t) - w2);
		double s = t0, fs = eliminateInf(ROCsq(s) - w2);
		double r = s, fr;

		for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
			r = (fs * t - ft * s) / (fs - ft);
			fr = ROCsq(r) - w2;
			if (sameSign(fr, ft)) {
				ft = fr;
				t = r;
				if (side < 0) {
					fs /= (1 << (-side));
					side--;
				} else {
					side = -1;
				}
			} else if (fr * fs > 0.0d) {
				fs = fr;
				s = r;
				if (side > 0) {
					ft /= (1 << side);
					side++;
				} else {
					side = 1;
				}
			} else {
				break;
			}
		}
		return r;
	}

	private static boolean sameSign(final double x, final double y) {
		// another way is to test if x*y > 0. This is bad for small x, y.
		return (x < 0.0d && y < 0.0d) || (x > 0.0d && y > 0.0d);
	}

	// returns the radius of curvature squared at t of this curve
	// see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
	private double ROCsq(final double t) {
		final double dx = t * (t * dax + dbx) + cx;
		final double dy = t * (t * day + dby) + cy;
		final double ddx = 2.0d * dax * t + dbx;
		final double ddy = 2.0d * day * t + dby;
		final double dx2dy2 = dx * dx + dy * dy;
		final double ddx2ddy2 = ddx * ddx + ddy * ddy;
		final double ddxdxddydy = ddx * dx + ddy * dy;
		return dx2dy2 * ((dx2dy2 * dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy * ddxdxddydy));
	}
}
